This paper is devoted to the study of reproduction of asymptotic boundedness in the second moment and small moments of stochastic differential equations by the stochastic theta method. In addition, we illustrate that the asymptotic moment boundedness of the numerical solution stand-alone plays a key role in the study of numerical stationary distribution.

The numerical reproduction of asymptotic properties of stochastic differential equations (SDEs) studies that given the underlying SDE has certain asymptotic property how one chooses a proper numerical method such that the corresponding discrete numerical solution can reproduce the same property. Among different types of asymptotic properties, stability has been attracting a lot of attention in recent years. Many papers are devoted to the numerical reproduction of stability of SDEs in different senses, such as mean square stability [1–3], almost sure stability [4–6], stability in small moment [7], we just mention some of them here. Another asymptotic property, asymptotic boundedness, was studied rarely, but has its own interest. Not like the stability that requires the solution to tend to the trivial solution as time becomes large, the boundedness only needs the solution to be bounded above by some positive constant. On the one hand, the stability could be regarded as a specific situation of the boundedness. More importantly, the asymptotic boundedness plays an important role in the study of stationary distribution of SDEs. In the series papers of Mao, Yuan, Yin, etc. [8–11], the stationary distributions of numerical solutions were used to approximate the stationary distribution of underlying equations. One of the key components in proving the existence and uniqueness of the numerical stationary distribution is the moment boundedness of numerical solutions. We will give more details about it in Section 4.

In this paper, we investigate the asymptotic moment boundedness of the stochastic theta method (STM). As the parameter theta is employed to control the implicitness of the method, the STM is regarded a generalisation of the Euler-Maruyama (EM) method and the backward Euler-Maruyama (BEM) method. The stabilities in different senses of the STM have been studied by many authors [12–14]. But, to our best knowledge, few papers have discussed the asymptotic moment boundedness of the STM for SDEs. Recently, in [11], the authors studied the moment boundedness for the EM method and the BEM method. The results presented in this paper can be seen as a generalisation of those in [11]. In addition, different choices of the θ will lead to distinguishing conditions on the drift and diffusion coefficients. We will study the asymptotic boundedness in the second moment and p th moments for p much less than one. The study of the second moment is typical as it can be related to many concepts in engineering such as energy function. While the small moments may have no obvious physical meanings, but they can be connected to the boundedness in probability which is crucial to the proof of the numerical stationary distribution. Besides, for the small moments, the conditions required for the drift and diffusion coefficients are weaker than those for the second moment.

We construct this paper as follows. In Section 2, some mathematical preliminaries are stated. The main results and proofs are presented in Section 3. The application of the moment boundedness in the study of numerical stationary distribution is discussed in Section 4.

Throughout this paper, we let (Ω,F,{Ft}t≥0,P) be a complete probability space with a filtration {Ft}t≥0 which is increasing and right continuous, with F0 containing all ℙ-null sets. Let B(t) be a scalar Brownian motion defined on the probability space. The results in this paper could be extended to the case of multi-dimensional Brownian motion. But to keep the simplicity of the notation, we only consider the case of scalar Brownian motion. Let |⋅| denote the Euclidean norm in Rn. The inner product of x, y in Rn is denoted by 〈x,y〉. In this paper, we consider the n-dimensional Itô SDE

dx(t)=f(x(t))dt+g(x(t))dB(t),t≥0,x(0)∈Rn.

(2.1)

We assume that f,g:Rn→Rn are smooth enough for SDE (2.1) to have a unique global solution on [0,∞).

Let us recall the stochastic θ numerical methods we will use below. The reader is referred to [15–17] for more details on the numerical methods. The stochastic theta method (STM) applied to (2.1) is defined by

xk+1=xk+(1−θ)f(xk)Δt+θf(xk+1)Δt+g(xk)ΔBk,x0=x(0),

(2.2)

for k=0,1,… , where Δt is the time step and ΔBk=B((k+1)Δt)−B(kΔt) is the Brownian motion increment.

Since the STM is semi-implicit when θ≠0, to ensure that this method is well defined, let us impose the one-sided Lipschitz condition on the drift coefficient f: there exists a constant b such that for any x,y∈Rn,

〈x−y,f(x)−f(y)〉≤b|x−y|2.

This, together with θbΔ<1, ensures that (2.2) is well defined, that is, STM (2.2) can be solved uniquely for the next step xk+1 (see, for example, [4]).

Our main results and their proofs are presented in this section. We start off with the second moment in Section 3.1, two cases of the θ are discussed. Then the same structure is used for Section 3.2 to investigate the small moments.

3.1 The second moment

First we discuss the situation for θ∈[0,1/2), which has the linear growth condition on both drift and diffusion coefficients. Second we relax the constraint on the drift coefficient when θ∈[1/2,1].

The boundedness of the underlying SDE is well known. We state the following theorem and refer the readers to Chapter 5 of [18] for the proof.

Theorem 3.1Assume thatfandgsatisfy the local Lipschitz condition. Assume that there exists a negative constantμand positive constantsσ, a1, a2such that for anyx∈Rn,

〈x,f(x)〉≤μ|x|2+a1

(3.1)

and

|g(x)|2≤σ|x|2+a2.

(3.2)

If

2μ+σ<0,

(3.3)

then the underlying solution of SDE (2.1) is asymptotically bounded in the second moment

where mk=[|g(xk)|2(ΔBk2−Δt)+2〈xk+(1−θ)f(xk)Δt,g(xk)ΔBk〉]. Taking expectation on both sides, noting that E(mk)=0, yields

E|xk+1|2≤c1E|xk|2+c2≤c1k+1E|x0|2+c2(1−c1k+1)1−c1,

(3.7)

where

c1=1+(1−θ)2κΔt2+σΔt+2(1−θ)μΔt1+2μθΔt

and

c2=(1−θ)2a3Δt2+a2Δt+2a1Δt1+2μθΔt.

Then, for Δt<−4(2μ+σ)κ, we have c1<1. From (3.7), we deduce

lim supk→∞E|xk|2≤c21−c1≤2a1+a2+(1−θ)2a3Δt−(2μ+σ)−(1−θ)2κΔt.

Let Δt→0, then assertion (3.6) holds. □

This theorem shows that the STM can reproduce the upper bound of true solution (3.4) for the case of θ∈[0,1/2). The result of the EM boundedness, Theorem 5.2 in [19], is reproduced perfectly as a special case.

3.1.2 θ∈[1/2,1]

We try to release the constraint on the drift coefficient when θ∈[1/2,1] and reproduce the boundedness property in STM as well. To show the theorem of this case, we first present the following lemma.

Lemma 3.3Let conditions (3.1) and (3.3) hold, then for anyA,B∈RwithA≥B≥0, we have the inequality

where C<1, E|x0−Af(x0)Δt|2+2Aa1Δt+2μ(θ−A)E|x0|2Δt≥0, 2(θ−A)μE|xk+1|2Δt≤0. Noting that

ϕλ=4(1−θ)μ+σ+2Cμ(2θ−1−λ)

when θ≤1+σ/4μ, thus λ=2θ−1, we have ϕλ≤0 easily; when θ>1+σ/4μ, thus λ=(1+σ2μ)(1−2μΔt(1−θ)1+2μΔt(1−θ)+σΔt), we still have ϕλ≤0.

Let k→∞, we have

lim supk→∞E(|xk|2)≤(2a1+a2)Δt1−C≤(2a1+a2)(1−2μΔtA)−2μλ,∀x0∈Rn.

Let Δt→0, assertion (3.8) and the special case θ=1 hold. □

Without the linear growth condition on the drift coefficient, this theorem shows that the STM can still reproduce the boundedness property of true solution (3.4). The result of the BEM boundedness, Theorem 5.4 in [19], is recovered perfectly as a special case when θ=1.

3.2 The small moment

In this section, we discuss the asymptotic boundedness of STM in the p th moment for small p. First we discuss the situation for θ∈[0,1/2), which has a linear growth condition on both drift and diffusion coefficients. Second we release the constraint on the drift coefficient when θ∈[1/2,1].

3.2.1 θ∈[0,1/2)

We begin by imposing the linear growth condition on both drift and diffusion coefficients of SDE (2.1):

|f(x)|2∨|g(x)|2≤κ|x|2+a,∀x∈Rn,

(3.10)

where κ and a are positive constants. We first present the theorem on the asymptotic boundedness in small moment of the solution of (2.1).

Theorem 3.5Let (3.10) hold. If there exists a positive constantDsuch that for anyx∈Rn,

〈x,f(x)〉+12|g(x)|2D+|x|2−〈x,g(x)〉2(D+|x|2)2≤−λ+P3(|x|)(D+|x|2)2,

(3.11)

whereλis a positive constant andPi(|x|)is a polynomial of|x|with degreei, then there existsp∗∈(0,1)such that for all0<p<p∗the solution of (2.1) obeys

Following the same technique as the one used in Theorem 5.2 in [18], by choosing the Lyapunov function V=(D+|x(t)|2)p/2, it is straightforward to prove this theorem. So we omit it here. Now we give the result for the STM solution.

Theorem 3.6Let (3.10) and (3.11) hold, andλ>θ(1+κ). Then, for anyε∈(0,λ−θ(1+κ)), there exists a pair of constantsp∗∈(0,1)andΔt∗∈(0,1)such that for∀p∈(0,p∗)and∀Δt∈(0,Δt∗), the STM solution (2.2) satisfies

where C1′ is a positive constant dependent on κ and p, C2′ is a positive constant dependent on κ, a, p and D, and both of them may change from line to line. Taking expectations on both sides, we obtain

Due to θ(1+κ)−λ+ε<0, we have 1+p(θ(1+κ)−λ+ε)Δt<1. Then, by iteration and letting k→∞, we have

lim supk→∞E(|xk+1|p)≤lim supk→∞E((D+|xk+1|2)p/2)≤C2′p(λ−θ(1+κ)−ε).

□

The theorem shows that the STM can reproduce the boundedness property of true solution (3.4). The result of the EM boundedness, Theorem 3.2 in [19], is recovered as a special case when θ=0.

3.2.2 θ∈[1/2,1]

In this part, we consider the case θ∈[1/2,1]. One may notice from the next theorem that in this case the parameter θ exists in the conditions, therefore the boundedness of the underlying equation may not be fully reproduced under the same conditions. However, as we stated in Section 1 that the asymptotic moment boundedness of the numerical as a stand-alone result is a key component in the study of numerical stationary distribution. Thus we still keep the next theorem and the problem that if one could construct some θ independent sufficient conditions for this case remains open.

Theorem 3.7Assume that the drift coefficient satisfies (3.1) and the diffusion coefficient satisfies (3.10) if the following holds for some positive constantλ:

where c1′ is a positive constant dependent on κ and p, c2′ is a positive constant dependent on κ, a, p and D, and both of them may change from line to line. Taking expectations on both sides, we obtain

For any ε∈(0,λ+p−28(1+κ)(1/θ2+κ)), by choosing p∗ sufficiently small such that p∗κ(1−2μθΔt)≤ε, then choose Δt∗∈(0,1) sufficiently small for p∗Δt∗[λ+p−28(1+κ)(1/θ2+κ)]≤1 and c2′Δt∗≤12p∗ε. For any p∈(0,p∗) and any Δt∈(0,Δt∗), we have

In this section, we illustrate the application of the results in the last section to the study of numerical stationary distribution.

Recalling Theorem 3.1 in [10], the authors proved that for any given one-step numerical method if the following three assumptions hold and the numerical solution is a homogeneous Markov process with a proper transition probability kernel, then the numerical solution has a unique stationary distribution as time tends to infinity.

Assumption 4.1 For any ε>0 and x0∈Rd, there exists a constant R=R(ε,x0)>0 such that

P(|xkx0|≥R)<εfor any k≥0.

Assumption 4.2 For any ε>0 and any compact subset K of Rd, there exists a positive integer k∗=k∗(ε,K) such that

P(|xkx0−xky0|<ε)≥1−εfor any k≥k∗ and any (x0,y0)∈K×K.

Assumption 4.3 For any ε>0, n≥1 and any compact subset K of Rd, there exists R=R(ε,n,K)>0 such that

P(sup0≤k≤n|xkx0|≤R)>1−εfor any x0∈K.

It is clear that Assumption 4.1 is satisfied by the results in Section 3 and the Chebyshev inequality. Furthermore, it is not hard to see that one can adapt the proofs in the previous section to show that for p=2 and some small enough p, E|xkx0−xky0|p tends to 0 as time becomes large. Then Assumption 4.2 follows. Due to the page limit, we omit the proof here. Assumption 4.3 can be obtained from the finite time moment boundedness of the STM; see, for example, [15]. In addition, it is easy to adapt the proof of Theorem 2.7 in [11] to show that the numerical solution derived from STM is a homogeneous Markov process with a proper transition probability kernel.

Therefore, one can see that there exists a unique stationary distribution for the numerical solution generated by the STM. As stated in [8–11], the reason to study the numerical stationary distribution is to approximate the stationary distribution of the underlying equations by avoiding solving the nontrivial Kolmogorov-Fokker-Planck partial differential equation. A more interesting open problem to us is if the numerical stationary distribution could be used as numerical solutions to certain type of partial differential equations.

One may see that those three assumptions are given in the sense of probability, but the existing sufficient conditions for Assumptions 4.1 and 4.2 are all in moment. The small moments as illustrated in this paper need weaker conditions than the second moment, but those conditions are still much stronger than those for underlying SDEs [20, 21] in which the sufficient conditions are given in the format of Lyapunov V functions. Thus, another interesting open problem is if one can construct some sufficient conditions in the format of Lyapunov V so that the assumptions can be satisfied directly without via the moment results.

Acknowledgements

The authors would like to thank the National Natural Science Foundation of China (Grant No. 11471071) and the Natural Science Foundation of Shanghai (Grant No. 14ZR1401200) for their financial support.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have equal contributions to each part of this paper. All the authors read and approved the final manuscript.

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